The objective of the proposed collaborative research is to carry out a comprehensive mathematical investigation of the key processes governing biopreservation of cells by vitrification and desiccation. The research will be focused on the formation of a glassy state in biological solutions and the interaction of the propagating glass transition fronts with cell membranes and intracellular organelles. This will be done by formulating the corresponding mathematical models and their investigation and solution by means of asymptotic and numerical methods. The proposed research program is planned as a close collaboration between a theoretical group at Northwestern University and an experimental group at the Harvard Medical School and the Massachusetts General Hospital. The results of theoretical analysis will be compared, analyzed and supported by experiments. The peculiarity of glassy systems is that the diffusion transport there is anomalous in that it is described by subdiffusion equations with fractional derivatives. From fundamental point of view, this research will allow one to understand the mechanisms of biopreservation based on the formation of glass inside the cell. From the biomedical point of view, this understanding will lead to improvement and optimization of biopreservation techniques and protocols to achieve higher cell survival rates. From the applied mathematics point of view, this research will develop new models and methods of solving special classes of non-trivial fractional PDEs (integro-differential equations) and associated free-boundary problems. The planned experimental work will be used as a guide in formulating the mathematical models. Analytical solutions will be sought in several limiting cases where asymptotic and perturbation methods can be applied, while a computational investigation will be done in the more general cases.